Characterizations Of Some Finite Simple Groups By Their Noncommuting Graphs Or "Two Orders"

Generally, there is an intimate relation between the groups and graphs, and in many occasions properties of graphs give rise to some properties of groups and vice versa. For instance, the prime graph GK(G) associated with a finite group G introduced by Gruenberg and Kegel who give a classification of finite groups by the number of connected components of their prime graph(cf.[39]). After that, there are many articles which give the characterizations of simple groups from the different conditions such as the conditions" the order of group and the set of element orders " and " the set of element orders " (cf. [3, 5, 12, 14, 20, 25, 26, 27, 28, 29, 31, 32, 33, 34, 38, 40]).One of these graphs that has attracted the attention of many authors is the non-commuting graph (?)(G) associated with a finite group G. It has been defined in [22] as follows: the vertex set of (?)(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity.In 1987, Professor J.G. Thompson put forward the following conjecture:Thompson' s Conjecture If G is a finite centerless group and M is a non-abelian simple group such that N(G) = N(M), then G≌M, where N(G) := {n∈N·| G has a conjugacy class of size n}.The validity of Thompson' s conjecture is known for all finite non-abelian simple groups whose prime graph are disconnected and we can refer to [8, 9, 10, 11].The validity of Thompson' s conjecture is not known for finite non-abelian simple groups with connected prime graphs.It has been the motivation of many research articles to study the finite sinple group by its non-commuting graph.In 2005, A.R. Moghaddamfar, W.J. Shi, W. Zhou and A.R. Zokayi proved if there exists a finite group whose noncommuting graph is isomorphic to one of the following groups: A_n,S_n, sporadic simple groups and Lie type simple groups with disconnected prime graph, their orders are equal (cf. [22]).In 2006, A. Abdollahi, S. Akbari and H.R. Maimani proved if there exists a finite group whose noncommuting graph is isomorphic to one of the following groups: PSL(2,2ⁿ),Sz(2^2m+1), they are isomorphic and put forword the following conjecture(cf.[1]):AAM's Conjecture Let M be a finite simple group and G a finite group satisfying (?)(G)≌V(M), then G≌M.The author considers AAM' s conjecture, discusses the groups L₂(q),L₃(q), finite simple groups with disconnected prime graph and the alternating group A₁₀, of degree 10 with connected prime graph and proves the following theorems in Chapter 2: Theorem 2.2.4 Let G be a finite group such that (?)(G)≌(?)(M), where M = L₂(q), then G≌M.Theorem 2.3.5 Let G be a finite group such that (?)(G)≌(?)(M), where M = L₃(q), then G≌M.Theorem 2.4.3 Let M be a finite simple groups with disconnected prime graph, G be a finite group, satisfying (?)(G)≌(?)(M), then G≌M.Theorem 2.5.7 Let G be a finite group such that (?)(G)≌(?)(A₁₀), then G≌A₁₀.The order of group G and the element order which are important in the theory of finite groups especially in the finite non-ablian simple groups are the basic arithmetic. Let Gbea finite group. Denote byÏ€(G) the set of prime divisors of |G|, byÏ€_e(G) the set of elements orders in G and N(G) the set of conjugacy sizes in G.In [39], the author gives the definition of prime graph GK(G) of G whose vertex is V(GK(G)) =Ï€(G) and edge E(GK(G)) = {p-q| pq∈π_e(G),p,q∈V(GK(G))}.In 1987, Professor Shi Wujie put forward the following conjecture:Conjecture Let G be a finite group and M a finite nonabelian simple group. Then G≌M if and only if (1)Ï€_e(G) =Ï€_e(M), (2) |G| = |M|.The author considers the above conjecture and proves the following Theorem 3.2.8 in Chapter 3:Theorem 3.2.8 Let Gbea finite group and M = D_n(2), where n is even. Then G≌M if and only if (1)Ï€_e(G) =Ï€_e(M), (2) |G| = |M|.The above theorem is complementary to unsolved cases in [40].For each finite simple group G of Lie type of characteristic p, define a prime power graphΓ(G) as follows in [19]. The vertices ofΓ(G) are the prime powers r^a that occur as orders of some elements of G, for all primes r≠p and integers a > 0. Prime powers r^a, s^b are adjacent if and only if G has an element of order lcm(r^a, s^b).In Chapter 4, We investigate the connection of prime power graph T(G) of simple group G of Lie type of characteristic p and give a classification of simple group of Lie type with complete graph components of T(G) and get the following results:Theorem 4.2.5 The number of components inΓ(G) of simple group G of Lie type is at most five.Theorem 4.3.6 Let G be a finite simple group of Lie type, and let all connected components of its prime power graphΓ(G) be cliques. Then G is one of the groups in the following list:(1) A₁(q), where q > 3;(3)A₂(q),where (3,q-1)=1; (4)A₃(2);(5)²A₂(q),where (3,q+1)=1;(6)C₂(q), where g> 2;(7)²B₂(q), where q = 2^2k+1;(8)G=G₂(q),where q=3^k.